Spin transport in non-Hermitian quantum systems

Transport in non-Hermitian quantum systems is studied. The goal is a better understanding of transport in non-Hermitian systems like the Lieb lattice due to its flat bands and the integrability of the Ising chain which allows transport in that model to be computed analytically. This is a very special feature that is not present in a generic non-Hermitian system. We obtain the behaviour of the spin conductivity as a function of the non-Hermitian parameters of each system with aim to verify the influence of variation them on conductivity. For all models analyzed: Ising model as well as noninteracting fermion models, we obtain a little influence of the non-Hermitian parameters on conductivity and thus, a small effect over transport coefficients. Furthermore, we obtain an influence of opening of the gap in the spectrum in these models on longitudinal conductivity as well.


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The representation of the energy bands �(k) , on complex plane is displayed in Fig. 2. The behavior of �(k) induced by the coupling parameters of the non-Hermitian model will generate a large influence on continuum and DC conductivities.
� 1 (k) = 2 3 c(k) 2 + d(k) 2 cos θ 1 3 , j , c † j , c j are the creation and annihilation operators of spinless fermions. The new operators satisfy the fermionic anticommutation relation {ḡ j , g ′ j } = δ j,j ′ . We set {|ψ n �} as the eigenstates of the operator i σ z j that represents all possible spin configurations along the +z direction. To proceed, one introduces a similarity j , represents a counterclockwise spin rotation in the σ x − σ y plane around the σ z axis by an angle θ : θ = tan −1 (iγ ) , which is a complex number that depends on the strength of the complex field. Under the biorthogonal basis of {A −1 j |ψ n �} and {A † j |ψ n �} , the matrix form of H is Hermitian for |δ| < 1 . The parity of the number of fermions is a conservative quantity such that the Hamiltonian can be expressed as H =H + I =H − I , where H + =H − = −2(ḡ Nḡ1 +ḡ N g 1 +ḡ 1 g N + g 1 g N ) and the Hamiltonian is rewritten as Taking the discrete Fourier transform where |k| = k = 2π(n + 1/2)/N , n = 0, 1, 2, ..., N − 1 , the Hamiltonian can be written as . The Hamiltonian is recast in the diagonal form with the dispersion relation of quasi-particles given by . www.nature.com/scientificreports/ If |δ| < 1 , the single-particle energy is real and |δ| > 1 , the system presents a complex single-particle spectrum regardless of k.

Transport
In the linear response theory for Hermitian systems, the response of the system to the frequency-dependent gradient of the external magnetic field h generates a spin current given by J = σ ∇h , where the response linear to the external field in x direction is being the response function defined as where is the Heaviside step function. On the other hand, the non-Hermitian response function is given by 43 where {· · ·} is the unequal-time anti-commutator to establish the link between the response function and the correlation function. We have the non-Hermitian dynamic susceptibility as the Fourier transform where τ = it . χ jS is the non-Hermitian response function The wave-vector-dependent susceptibility is given by From continuity for the spin current: S z (k, t) + ik · J x (k, t) = 0 , χ NH jS can be transformed as follows: Using the representation of the spin current operator in terms of spin operators where j + x is the nearest-neighbor site of the site j in the positive x direction, one can transform the second term as In the long-wavelength k x → 0 limit the susceptibility χ NH jS (k, ω) is thus proportional to ik x and we can write where �−K x � is the kinetic energy, being given by www.nature.com/scientificreports/ and G is the Green's function defined in T = 0 by 44 being T , the time ordering operator. The regular part of the conductivity σ (continuum conductivity) in the context of Hermitian quantum mechanics is given by [44][45][46][47][48][49] : Re[σ (ω)] = D S (T)δ(ω) + σ reg (ω) , where and α, β = x, y, z . D S (T) is the spin Drude's weight, being given by where n(ω k ) = 1/(e βω k ± 1) is the occupation number of bosons and fermions and β = 1/T.
The behavior of Drude's weight D S (T) as a function of T is displayed in Fig. 3 for the Ising model Eq. (13). The effective T that best relates the susceptibilities via fluctuation dissipation relation for a fixed waiting time t w is given as 43 where χ NH = χ ′NH + iχ ′′NH , χ = χ ′ + iχ ′′ . For δ = 0 we have the Hermitian model and δ = 0 the model is non-Hermitian. We obtain a small difference in the behavior of the curves for the two models (Hermitian and non-Hermitian) due to transformation of the non-Hermitian Hamiltonian in Hermitian, Eq. (14). Moreover, for T non-zero, D S (T) rises with T however, this description is only qualitative due to approach used.
The continuum part of the spin conductivity σ reg (ω) , is defined in terms of the Green's function G(ω).
We obtain the spin current operator in terms of the operators ψ † and ψ given by The spin current response function G(k, ω) at non-zero T is given by 44 where G(k = 0, ω → 0) is the susceptibility or retarded Green's function 44 . The retarded Green's function or dynamical correlation function is obtained after performing an analytical calculation, where we obtain the result N k (ω) is the Fourier transform of N k (t) , which is the dynamical correlation function Consequently, we obtain the regular part of the longitudinal spin conductivity σ reg (ω) as being given by In all cases analyzed, the influence of dispersionless flat modes on longitudinal spin conductivity is only to give rise to a Dirac's delta-like peak at frequency ω = ω k , where ω k is a plane mode in each case. Furthermore, the presence of large peaks in the AC spin conductivity and a finite Drude's weight D S (T) , indicate a supercurrent behavior for the system although, for one has a superconductor behavior is necessary that the system exhibits the Meissner effect as well 50 .
In Fig. 4, we present the behavior of σ reg (ω) for different values of non-Hermitian coupling δ . We obtain the AC conductivity tending to zero at ω → 0 however, as we have σ (0) = D S δ(ω) and since that we obtain a D S finite, we must have a divergence for the DC current. However, the scattering among particles must introduce a spreading in the conductivity where in a real system the conductivity must to stay finite. The large peaks obtained for the conductivity are due to the behavior of the dispersion relation at range (1.0 < ω/J < 3.0) , generating so, resonance effects on conductivity. In Figs. 5 and 6, we analyze the conductivity for the non-Hermitian model Eq.
(2). In this case, one obtains a divergence in the continuum conductivity at DC limit, ω → 0 . The behavior obtained for the AC conductivity is due to the form of the Eqs. (11) and (43), which are very complicated expressions of k , involving thus, many processes that depends on k . For the Hermitian model on Lieb lattice, we must have the canceling of some terms in Eq. (4) however, the expression for σ reg (ω) does not change a lot and hence, the behavior for the conductivity at ω → 0 must be the same. Furthermore, as we obtain a finite Drude's weight for all values of T, we have a Dirac's delta peak for the conductivity at ω = 0 and consequently, we obtain that the For δ = 0 we have the Hermitian model and δ = 0 , the model is non-Hermitian. We find that conductivity tends to zero at DC limit. We make = 1.0 in the calculations. www.nature.com/scientificreports/ transport is ideal in this point ( ω = 0 ) for all values of T. For values nonzero of ω ( ω = 0 ), we obtain a decreasing in the conductivity for higher values of T and ω , although this behavior is only qualitative due to approach used.

Summary
In brief, we analyze the transport for the 2D non-Hermitian Lieb lattice and Ising model which are important models of quantum dissipative systems. The analysis for the XXZ model may be made in a future work. As far as I know, there is none experimental result that investigates the influence of energy bands on spin conductivity for the non-Hermitian models considered here. However, the rapid advance of experimental techniques in the last years has allowed the study of many systems in more complex lattices geometries 41,[51][52][53][54][55][56][57] . In a general way, in quantum spin systems, either real fields or complex fields generate a splitting of the degenerate ground states, where the spins are aligned along of the direction of the external magnetic field. The eigenvalues and the eigenvectors of the system with real spectrum do not change with the external magnetic field and in general, the initial state display a oscillating behavior and periodic among all possible spin configurations. This situation change a lot when a critical complex field is applied. The eigenstates and the dynamical behavior suffer a large change where all the initial states evolve to a coalescent state independent of the initial spin configurations. Thus, it is interesting to obtain the intriguing features of a quantum spin system in the presence of complex fields.

Data availibility
All data generated or analysed during this study are included in this paper. (1). We obtain the conductivity tending to the infinity at DC limit, indicating thus an ideal transport in this limit. www.nature.com/scientificreports/